λx.x denotes the identity function, which simply returns its argument;
λx.c denotes the constant function, which always returns the constant c regardless of its argument;
λx.f(f(x)) denotes the composition of the function f with itself, i.e. the function that, for any argument x, returns f(f(x)).
Much of the power of the notation derives from the ability to represent higher-order functions. For example, λf.λx.f(f(x))
denotes the (higher-order) function that, when applied to a function f, returns the function obtained by composing f with itself.
As well as a notation, the λ-calculus comprises rules for reducing λ-expressions to equivalent ones. The most important is the rule of β-reduction, by which an expression of the form (λx.e1)(e2)
reduces to e1 with all free occurrences of x replaced by e2. For example, (λx.f(λx.x,x))(a)
reduces to f(λx.x,a)
As a second example, involving a functional variable, the expression (λf.f(a))(λx.g)(x,b))
reduces to (λx.g(x,b))(a)
and hence to g(a,b)
In theoretical terms, the formalism of λ-calculus can be shown to be equivalent in expressive power to that of Turing machines. It has a special role in the study of programming languages: one can point to its influence on the design of functional languages such as J. McCarthy's LISP; to P. Landin's reduction of Algol 60 to λ-calculus, and to D. Scott's construction of a set-theory meaning for the full unrestricted λ-calculus – a construction that ushered in the theory of domains in the denotational semantics of programming languages.
"lambda calculus." A Dictionary of Computing. . Encyclopedia.com. (August 19, 2018). http://www.encyclopedia.com/computing/dictionaries-thesauruses-pictures-and-press-releases/lambda-calculus
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